1st character - Get to 340 (345 with full +5 mods), then do 2-3 milestones (or 1 if you're super hardcore and want to grind a lot of blues/vendor packages). After that, grind out blue drops until you can't climb higher, then do more milestones and grind again. When out of milestones, run the normal Leviathan, raid lair and then prestige Leviathan. Transfer (hopefully) high level weapons to next characters.
2nd and 3rd characters - Finish campaign (I guess?) and then try to run prestige raid. Prestige raid only drops armor, so you have no risk of wasting a drop getting a high level weapon which wouldn't improve your power level a lot, whereas the armor drops will. After prestige raid, follow similar pattern to first character, milestone or two, then grind, then milestone, etc, etc. Transfer weapons to 3rd character and repeat.